Hom-约当三系的表示与扩张

Representation and Extension of Hom-Jordan Triple System

摘要: 本文主要介绍了Hom-约当三系的表示与扩张。首先，给出了Hom-约当三系表示的定义、判断表示的方法以及表示的例子，同时给出Hom-约当三系的表示的对偶是表示所满足的条件。其次，找到Hom-约当三系扩张的两种方法，其一是利用一个特殊的线性函数而构造出的扩张，其二是Hom-约当三系的T^*-扩张。

Abstract: This paper mainly studies the representation and extension of Hom-Jordan triple system. Firstly, we give the definition of representation of Hom-Jordan triple system, the method of judging the representation and an example of representation. At the same time, the condition that the dual of Hom-Jordan triple system is the representation is given. Secondly, two methods of extension of Hom-Jordan triple system are found. One is to construct an extension by a special linear function, and the other is the T^*-extension of Hom-Jordan triple system.

1. 引言

约当代数和约当三系与对称流形有密切联系，实的紧的约当三系与对称R-空间之间存在一一对应关系 [1]。此外，约当三系与结合代数也有密切联系。在结合代数 $\left(A,\circ \right)$ 上定义运算 $a\circ \left(b\circ c\right)+b\circ \left(a\circ c\right)+\left(a\circ b\right)\circ c$ 可以得到约当三系 [2]。约当三系作为独立的代数体系已经有很多相关的结果，例如约当三系的分类 [3]、约当三系的表示 [4]、约当三系的上同调 [5] 等。Hom-约当三系是 [6] 中引入的，是约当三系的推广。本文将考虑Hom-约当三系的表示和扩张。

2. Hom-约当三系的表示

本文所指的线性空间都是数域K上的有限维线性空间。

定义1.1 [5] 设J是线性空间， ${\alpha }_1,{\alpha }_2\in End\left(J\right)$， $\left\{\cdot ,\cdot ,\cdot \right\}:J\times J\times J\to J$ 是三线性映射，如果对于任意的 $x,y,z,u,v\in J$ 满足下列条件

$\left\{x,y,z\right\}=\left\{z,y,x\right\},$ (1.1)

$\begin{array}{l}\left\{{\alpha }_1\left(x\right),{\alpha }_2\left(y\right),\left\{z,u,v\right\}\right\}-\left\{{\alpha }_1\left(z\right),{\alpha }_2\left(u\right),\left\{x,y,v\right\}\right\}\\ =\left\{\left\{x,y,z\right\},{\alpha }_1\left(u\right),{\alpha }_2\left(v\right)\right\}-\left\{{\alpha }_1\left(z\right),\left\{y,x,u\right\},{\alpha }_2\left(v\right)\right\},\end{array}$ (1.2)

则称 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系。

当 ${\alpha }_1=id,{\alpha }_2=id$ 时， $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为约当三系。

定义1.2设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系，V是线性空间， $A_1,A_2\in End\left(V\right)$， ${\theta }_{12},{\theta }_{13}:J\times J\to End\left(V\right)$ 为双线性映射，如果对于任意的 $x,y,z,u\in J$ 有

${\theta }_{13}\left(x,y\right)={\theta }_{13}\left(y,x\right),$ (1.3)

$\begin{array}{l}{\theta }_{12}\left({\alpha }_2\left(x\right),{\alpha }_1\left(u\right)\right){\theta }_{12}\left(z,y\right)-{\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(x\right)\right){\theta }_{13}\left(y,u\right)\\ -{\theta }_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right)A_1+{\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{12}\left(x,y\right)=0,\end{array}$ (1.4)

$\begin{array}{l}{\theta }_{12}\left({\alpha }_2\left(y\right),{\alpha }_1\left(u\right)\right){\theta }_{13}\left(x,z\right)-{\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(y\right)\right){\theta }_{12}\left(u,x\right)\\ -{\theta }_{13}\left({\alpha }_1\left(x\right),\left\{z,u,y\right\}\right)A_2+{\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{13}\left(x,y\right)=0,\end{array}$ (1.5)

$\begin{array}{l}{\theta }_{12}\left({\alpha }_2\left(z\right),{\alpha }_1\left(u\right)\right){\theta }_{12}\left(x,y\right)-{\theta }_{12}\left({\alpha }_2\left(z\right),\left\{y,x,u\right\}\right)A_1\\ -{\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){\theta }_{12}\left(z,u\right)+{\theta }_{12}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right)A_1=0,\end{array}$ (1.6)

$\begin{array}{l}{\theta }_{13}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right)A_1-{\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{12}\left(y,x\right)\\ -{\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){\theta }_{13}\left(z,u\right)+{\theta }_{13}\left({\alpha }_1\left(z\right),\left\{x,y,u\right\}\right)A_2=0,\end{array}$ (1.7)

$\begin{array}{l}{\theta }_{12}\left(\left\{x,y,z\right\},{\alpha }_1\left(u\right)\right)A_2-{\theta }_{12}\left({\alpha }_1\left(z\right),\left\{y,x,u\right\}\right)A_2\\ -{\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){\theta }_{12}\left(z,u\right)+{\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{12}\left(x,y\right)=0,\end{array}$ (1.8)

则称 $\left(V,{\theta }_{12},{\theta }_{13},A_1,A_2\right)$ 是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示。

例1.1设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系， $L:J\times J\to End\left(J\right)$， $M:J\times J\to End\left(J\right)$ 为双线性映射，其中 $L\left(x,y\right)\left(z\right)=\left\{x,y,z\right\}$， $M\left(x,y\right)\left(z\right)=\left\{x,z,y\right\}$ $\left(\forall x,y,z\in J\right)$，则 $\left(J,L,M,{\alpha }_1,{\alpha }_2\right)$ 为 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示，称为伴随表示。

证明：直接验证可知对于 $\left(J,L,M,{\alpha }_1,{\alpha }_2\right)$ 有(1.3)~(1.8)成立。

定理1.1设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系，V是线性空间， $A_1,A_2\in End\left(V\right)$， ${\theta }_{12},{\theta }_{13}:J\times J\to End\left(V\right)$ 为双线性映射，在 $J\oplus V$ 上定义

$\left\{x+a,y+b,z+c\right\}=\left\{x,y,z\right\}+{\theta }_{12}\left(z,y\right)a+{\theta }_{13}\left(x,z\right)b+{\theta }_{12}\left(x,y\right)c,$

$\left({\alpha }_1+A_1\right)\left(x+a\right)={\alpha }_1\left(x\right)+A_1\left(a\right),$

$\left({\alpha }_2+A_2\right)\left(x+a\right)={\alpha }_2\left(x\right)+A_2\left(a\right),$

其中 $\forall x,y,z\in J,\forall a,b,c\in V$，则 $\left(J\oplus V，\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1+A_1,{\alpha }_2+A_2\right)$ 为Hom-约当三系当且仅当 $\left(V,{\theta }_{12},{\theta }_{13},A_1,A_2\right)$ 是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示。

证明: $\left(J\oplus V,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1+A_1,{\alpha }_2+A_2\right)$ 为Hom-约当三系当且仅当下列等式成立

$\left\{x+a,y+b,z+c\right\}=\left\{z+c,y+b,x+a\right\},$ (1.9)

$\begin{array}{l}\left\{\left\{x+a,y+b,z+c\right\},\left({\alpha }_1+A_1\right)\left(u+d\right),\left({\alpha }_2+A_2\right)\left(v+e\right)\right\}\\ -\left\{\left({\alpha }_1+A_1\right)\left(z+c\right),\left\{y+b,x+a,u+d\right\},\left({\alpha }_2+A_2\right)\left(v+e\right)\right\}\\ -\left\{\left({\alpha }_1+A_1\right)\left(x+a\right),\left({\alpha }_2+A_2\right)\left(y+b\right),\left\{z+c,u+d,v+e\right\}\right\}\\ +\left\{\left({\alpha }_1+A_1\right)\left(z+c\right),\left({\alpha }_2+A_2\right)\left(u+d\right),\left\{x+a,y+b,v+e\right\}\right\}=0,\end{array}$ (1.10)

其中 $\forall x+a,y+b,z+c,u+d,v+e\in J\oplus V$。

由于

$\begin{array}{l}\left\{x+a,y+b,z+c\right\}-\left\{z+c,y+b,x+a\right\}\\ =\left\{x,y,z\right\}-\left\{z,y,x\right\}+{\theta }_{12}\left(z,y\right)a-{\theta }_{12}\left(z,y\right)a+{\theta }_{13}\left(x,z\right)b\\ -{\theta }_{13}\left(z,x\right)b+{\theta }_{12}\left(x,y\right)c-{\theta }_{12}\left(x,y\right)c,\end{array}$

因此(1.9)成立当且仅当 ${\theta }_{13}$ 对称，即(1.3)式成立。

(1.10)式左边直接计算得

$\begin{array}{l}\left\{\left\{x+a,y+b,z+c\right\},\left({\alpha }_1+A_1\right)\left(u+d\right),\left({\alpha }_2+A_2\right)\left(v+e\right)\right\}\\ -\left\{\left({\alpha }_1+A_1\right)\left(z+c\right),\left\{y+b,x+a,u+d\right\},\left({\alpha }_2+A_2\right)\left(v+e\right)\right\}\\ -\left\{\left({\alpha }_1+A_1\right)\left(x+a\right),\left({\alpha }_2+A_2\right)\left(y+b\right),\left\{z+c,u+d,v+e\right\}\right\}\\ +\left\{\left({\alpha }_1+A_1\right)\left(z+c\right),\left({\alpha }_2+A_2\right)\left(u+d\right),\left\{x+a,y+b,v+e\right\}\right\}\end{array}$

$\begin{array}{l}=\left\{\left\{x,y,z\right\},{\alpha }_1\left(u\right),{\alpha }_2\left(v\right)\right\}-\left\{{\alpha }_1\left(z\right),\left\{y,x,u\right\},{\alpha }_2\left(v\right)\right\}\\ -\left\{{\alpha }_1\left(x\right),{\alpha }_2\left(y\right),\left\{z,u,v\right\}\right\}+\left\{{\alpha }_1\left(z\right),{\alpha }_2\left(u\right),\left\{x,y,v\right\}\right\}\\ +({\theta }_{12}\left({\alpha }_2\left(v\right),{\alpha }_1\left(u\right)\right){\theta }_{12}\left(z,y\right)-{\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(v\right)\right){\theta }_{13}\left(y,u\right)\\ -{\theta }_{12}\left(\left\{z,u,v\right\},{\alpha }_2\left(y\right)\right)A_1+{\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{12}\left(v,y\right))\left(a\right)\\ +({\theta }_{12}\left({\alpha }_2\left(v\right),{\alpha }_1\left(u\right)\right){\theta }_{13}\left(x,z\right)-{\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(v\right)\right){\theta }_{12}\left(u,x\right)\end{array}$

$\begin{array}{l}-{\theta }_{13}\left({\alpha }_1\left(x\right),\left\{z,u,v\right\}\right)A_2+{\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{13}\left(x,v\right))\left(b\right)\\ +({\theta }_{12}\left({\alpha }_2\left(v\right),{\alpha }_1\left(u\right)\right){\theta }_{12}\left(x,y\right)-{\theta }_{12}\left({\alpha }_2\left(v\right),\left\{y,x,u\right\}\right)A_1\\ -{\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){\theta }_{12}\left(v,u\right)+{\theta }_{12}\left(\left\{x,y,v\right\},{\alpha }_2\left(u\right)\right)A_1)\left(c\right)\\ +({\theta }_{13}\left(\left\{x,y,z\right\},{\alpha }_2\left(v\right)\right)A_1-{\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(v\right)\right){\theta }_{12}\left(y,x\right)\end{array}$

$\begin{array}{l}-{\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){\theta }_{13}\left(z,v\right)+{\theta }_{13}\left({\alpha }_1\left(z\right),\left\{x,y,v\right\}\right)A_2)\left(d\right)\\ +({\theta }_{12}\left(\left\{x,y,z\right\},{\alpha }_1\left(u\right)\right)A_2-{\theta }_{12}\left({\alpha }_1\left(z\right),\left\{y,x,u\right\}\right)A_2\\ -{\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){\theta }_{12}\left(z,u\right)+{\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{12}\left(x,y\right))(\; e\; )\end{array}$

由 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系，则等式(1.2)成立。因此，(1.10)式成立当且仅当(1.4)~(1.8)成立。因此，结论成立。

设 $V,W$ 为线性空间， $\phi :V\to W$ 为线性映射，定义 ${\phi }^{\text{*}}:W^{*}\to V^{*}$，其中

$\langle {\phi }^{\text{*}}\left(f\right),v\rangle =\langle f,\phi \left(v\right)\rangle \left(\forall f\in W^{*},v\in V\right),$

则 ${\phi }^{\text{*}}$ 为线性映射，称为 $\phi$ 的对偶映射。

设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系，V为线性空间， $\theta :J\times J\to End\left(V\right)$ 为线性映射，定义 ${\theta }^{*}:J\times J\to End\left(V^{*}\right)$，其中

$\langle {\theta }^{*}\left(x,y\right)f,v\rangle =\langle f,\theta \left(x,y\right)v\rangle \left(\forall x,y\in J,v\in V,f\in V^{*}\right),$

则 ${\theta }^{*}$ 为线性映射。

定理1.2设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系， $\left(V,{\theta }_{12},{\theta }_{13},A_1,A_2\right)$ 是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示，则 $\left(V^{*},{\theta }_{12}^{*},{\theta }_{13}^{*},A_1^{*},A_2^{*}\right)$ 也是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示当且仅当满足下列条件

$\begin{array}{l}{\theta }_{12}\left(z,y\right){\theta }_{12}\left({\alpha }_2\left(x\right),{\alpha }_1\left(u\right)\right)-{\theta }_{13}\left(y,u\right){\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(x\right)\right)\\ -A_1{\theta }_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right)+{\theta }_{12}\left(x,y\right){\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)=0,\end{array}$ (1.11)

$\begin{array}{l}{\theta }_{13}\left(x,z\right){\theta }_{12}\left({\alpha }_2\left(y\right),{\alpha }_1\left(u\right)\right)-{\theta }_{12}\left(u,x\right){\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(y\right)\right)\\ -A_2{\theta }_{13}\left({\alpha }_1\left(x\right),\left\{z,u,y\right\}\right)+{\theta }_{13}\left(x,y\right){\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)=0,\end{array}$ (1.12)

$\begin{array}{l}{\theta }_{12}\left(x,y\right){\theta }_{12}\left({\alpha }_2\left(z\right),{\alpha }_1\left(u\right)\right)-A_1{\theta }_{12}\left({\alpha }_2\left(z\right),\left\{y,x,u\right\}\right)\\ -{\theta }_{12}\left(z,u\right){\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)+A_1{\theta }_{12}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right)=0,\end{array}$ (1.13)

$\begin{array}{l}{A}_1{\theta }_{13}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right)-{\theta }_{12}\left(y,x\right){\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\\ -{\theta }_{13}\left(z,u\right){\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)+A_2{\theta }_{13}\left({\alpha }_1\left(z\right),\left\{x,y,u\right\}\right)=0,\end{array}$ (1.14)

$\begin{array}{l}{A}_2{\theta }_{12}\left(\left\{x,y,z\right\},{\alpha }_1\left(u\right)\right)-A_2{\theta }_{12}\left({\alpha }_1\left(z\right),\left\{y,x,u\right\}\right)\\ -{\theta }_{12}\left(z,u\right){\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)+{\theta }_{12}\left(x,y\right){\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)=0,\end{array}$ (1.15)

其中 $\forall x,y,z,u\in J$。

证明； $\left(V^{*},{\theta }_{12}^{*},{\theta }_{13}^{*},A_1^{*},A_2^{*}\right)$ 为 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示当且仅当 $\left(V^{*},{\theta }_{12}^{*},{\theta }_{13}^{*},A_1^{*},A_2^{*}\right)$ 满足等式(1.3)~(1.8)。 $\forall x,y,z,u\in J,f\in V^{*},v\in V$，直接计算得

$\langle \left[{\theta }_{13}^{\text{*}}\left(x,y\right)-{\theta }_{13}^{*}\left(y,x\right)\right]f,v\rangle =\langle f,\left[{\theta }_{13}\left(x,y\right)-{\theta }_{13}\left(y,x\right)\right]v\rangle ,$

$\begin{array}{l}\langle [{\theta }_{12}^{*}\left({\alpha }_2\left(x\right),{\alpha }_1\left(u\right)\right){\theta }_{12}^{*}\left(z,y\right)-{\theta }_{13}^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(x\right)\right){\theta }_{13}^{*}\left(y,u\right)\\ -{\theta }_{12}^{*}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right)A_1{}^{*}+{\theta }_{12}^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{12}^{*}\left(x,y\right)]f,v\rangle \\ =\langle f,[{\theta }_{12}\left(z,y\right){\theta }_{12}\left({\alpha }_2\left(x\right),{\alpha }_1\left(u\right)\right)-{\theta }_{13}\left(y,u\right){\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(x\right)\right)\\ -A_1{\theta }_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right)+{\theta }_{12}\left(x,y\right){\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)]v\rangle ,\end{array}$

$\begin{array}{l}\langle [{\theta }_{12}^{\text{*}}\left({\alpha }_2\left(y\right),{\alpha }_1\left(u\right)\right){\theta }_{13}^{\text{*}}(x,z)-{\theta }_{13}^{\text{*}}\left({\alpha }_1\left(z\right),{\alpha }_2\left(y\right)\right){\theta }_{12}^{\text{*}}\left(u,x\right)\\ -\left({\alpha }_1\left(x\right),\left\{z,u,y\right\}\right)A_2^{\text{*}}+{\theta }_{12}^{\text{*}}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{13}^{\text{*}}\left(x,y\right)]f,v\rangle \\ =\langle f,[{\theta }_{13}\left(x,z\right){\theta }_{12}\left({\alpha }_2\left(y\right),{\alpha }_1\left(u\right)\right)-{\theta }_{12}\left(u,x\right){\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(y\right)\right)\\ -A_2\left({\alpha }_1\left(x\right),\left\{z,u,y\right\}\right)+{\theta }_{13}\left(x,y\right){\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)]v\rangle ,\end{array}$

$\begin{array}{l}\langle [{\theta }_{12}^{\text{*}}\left({\alpha }_2\left(z\right),{\alpha }_1\left(u\right)\right){\theta }_{12}^{\text{*}}\left(x,y\right)-{\theta }_{12}^{\text{*}}\left({\alpha }_2\left(z\right),\left\{y,x,u\right\}\right)A_1^{\text{*}}\\ -{\theta }_{12}^{\text{*}}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){\theta }_{12}^{\text{*}}\left(z,u\right)+{\theta }_{12}^{\text{*}}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right)A_1{}^{\text{*}}]f,v\rangle \\ =\langle f,[{\theta }_{12}\left(x,y\right){\theta }_{12}\left({\alpha }_2\left(z\right),{\alpha }_1\left(u\right)\right)-A_1{\theta }_{12}\left({\alpha }_2\left(z\right),\left\{y,x,u\right\}\right)\\ -{\theta }_{12}\left(z,u\right){\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)+A_1{\theta }_{12}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right)]v\rangle ,\end{array}$

$\begin{array}{l}\langle [{\theta }_{13}^{\text{*}}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right)A_1{}^{\text{*}}-{\theta }_{13}^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{12}^{*}\left(y,x\right)\\ -{\theta }_{12}^{*}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){\theta }_{13}^{*}\left(z,u\right)+{\theta }_{13}^{*}\left({\alpha }_1\left(z\right),\left\{x,y,u\right\}\right)A_2^{*}]f,v\rangle \\ =\langle f,[A_1{\theta }_{13}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right)-{\theta }_{12}\left(y,x\right){\theta }_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\\ -{\theta }_{13}\left(z,u\right){\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)+A_2{\theta }_{13}\left({\alpha }_1\left(z\right),\left\{x,y,u\right\}\right)]v\rangle ,\end{array}$

$\begin{array}{l}\langle [{\theta }_{12}^{\text{*}}\left(\left\{x,y,z\right\},{\alpha }_1\left(u\right)\right)A_2^{\ast }-{\theta }_{12}^{\text{*}}\left({\alpha }_1\left(z\right),\left\{y,x,u\right\}\right)A_2^{\ast }\\ -{\theta }_{12}^{*}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){\theta }_{12}^{*}\left(z,u\right)+{\theta }_{12}^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){\theta }_{12}^{*}\left(x,y\right)]f,v\rangle \\ =\langle f,[A_2{\theta }_{12}\left(\left\{x,y,z\right\},{\alpha }_1\left(u\right)\right)-A_2{\theta }_{12}\left({\alpha }_1\left(z\right),\left\{y,x,u\right\}\right)\\ -{\theta }_{12}\left(z,u\right){\theta }_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)+{\theta }_{12}\left(x,y\right){\theta }_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)]v\rangle ,\end{array}$

由 $\left(V,{\theta }_{12},{\theta }_{13},A_1,A_2\right)$ 是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示可知 ${\theta }_{13}\left(x,y\right)={\theta }_{13}\left(y,x\right)$，则 $\left(V^{*},{\theta }_{12}^{*},{\theta }_{13}^{*},A_1^{*},A_2^{*}\right)$ 上等式(1.3)~(1.8)成立等价于等式(1.11)~(1.15)成立。因此结论成立。

推论1.1设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系，则 $\left(J^{*},L^{*},M^{*},{\alpha }_1^{*},{\alpha }_2^{*}\right)$ 为 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示当且仅当J上的运算 $\left\{\cdot ,\cdot ,\cdot \right\}$ 满足下列条件

$\begin{array}{l}\left\{z,y,\left\{{\alpha }_2\left(x\right),{\alpha }_1\left(u\right),v\right\}\right\}-\left\{y,\left\{{\alpha }_1\left(z\right),v,{\alpha }_2\left(x\right)\right\},u\right\}\\ -{\alpha }_1\left(\left\{\left\{z,u,x\right\},{\alpha }_2\left(y\right),v\right\}\right)+\left\{x,y,\left\{{\alpha }_1\left(z\right),{\alpha }_2\left(u\right),v\right\}\right\}=0,\end{array}$ (1.16)

$\begin{array}{l}\left\{x,\left\{{\alpha }_2\left(y\right),{\alpha }_1\left(u\right),v\right\},z\right\}-\left\{u,x,\left\{{\alpha }_1\left(z\right),v,{\alpha }_2\left(y\right)\right\}\right\}\\ -{\alpha }_2\left(\left\{{\alpha }_1\left(x\right),v,\left\{z,u,y\right\}\right\}\right)+\left\{x,\left\{{\alpha }_1\left(z\right),{\alpha }_2\left(u\right),v\right\},y\right\}=0,\end{array}$ (1.17)

$\begin{array}{l}\left\{x,y,\left\{{\alpha }_2\left(z\right),{\alpha }_1\left(u\right),v\right\}\right\}-{\alpha }_1\left(\left\{{\alpha }_2\left(z\right),\left\{y,x,u\right\},v\right\}\right)\\ -\left\{z,u,\left\{{\alpha }_1\left(x\right),{\alpha }_2\left(y\right),v\right\}\right\}+{\alpha }_1\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right),v\right)=0,\end{array}$ (1.18)

$\begin{array}{l}{\alpha }_1\left(\left\{x,y,z\right\},v,{\alpha }_2\left(u\right)\right)-\left\{y,x,\left\{{\alpha }_1\left(z\right),v,{\alpha }_2\left(u\right)\right\}\right\}\\ -\left\{z,\left\{{\alpha }_1\left(x\right),{\alpha }_2\left(y\right),v\right\},u\right\}+{\alpha }_2\left(\left\{{\alpha }_1\left(z\right),v,\left\{x,y,u\right\}\right\}\right)=0,\end{array}$ (1.19)

$\begin{array}{l}{\alpha }_2\left(\left\{x,y,z\right\},{\alpha }_1\left(u\right),v\right)-{\alpha }_2\left(\left\{{\alpha }_1\left(z\right),\left\{y,x,u\right\},v\right\}\right)\\ -\left\{z,u,\left\{{\alpha }_1\left(x\right),{\alpha }_2\left(y\right),v\right\}\right\}+\left\{x,y,\left\{{\alpha }_1\left(z\right),{\alpha }_2\left(u\right),v\right\}\right\}=0,\end{array}$ (1.20)

其中 $\forall x,y,z,u,v\in J$。

证明：利用定理1.2，取 ${\theta }_{12}=L,{\theta }_{13}=M$，通过计算可直接得出。

设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系， $V,W$ 为线性空间， ${\theta }^{\prime }:J\times J\to End\left(V\right)$， ${\theta }^{″}:J\times J\to End\left(W\right)$ 为双线性映射， $A^{\prime }\in End\left(V\right)$， $A^{″}\in End\left(W\right)$，定义 ${\theta }^{\prime }\otimes {\theta }^{″}:J\times J\to End\left(V\otimes W\right)$， $A^{\prime }\otimes A^{″}\in End\left(V\otimes W\right)$，其中

${\theta }^{\prime }\otimes {\theta }^{″}\left(x,y\right)\left(v\otimes w\right)={\theta }^{\prime }\left(x,y\right)v\otimes w+v\otimes {\theta }^{″}\left(x,y\right)w,$

$A^{\prime }\otimes A^{″}\left(v\otimes w\right)=A^{\prime }\left(v\right)\otimes A^{″}\left(w\right),$

其中 $\forall x,y\in J,v\in V,w\in W$。

定理1.3设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系， $\left(V,{{\theta }^{\prime }}_{12},{{\theta }^{\prime }}_{13},{A^{\prime }}_1,{A^{\prime }}_2\right),\left(W,{{\theta }^{″}}_{12},{{\theta }^{″}}_{13},{A^{″}}_1,{A^{″}}_2\right)$ 均为 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示，则 $\left(V\otimes W,{{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12},{{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13},{A^{\prime }}_1\otimes {A^{″}}_1,{A^{\prime }}_2\otimes {A^{″}}_2\right)$ 是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示当且仅当 $\left(V,{{\theta }^{\prime }}_{12},{{\theta }^{\prime }}_{13},{A^{\prime }}_1,{A^{\prime }}_2\right),\left(W,{{\theta }^{″}}_{12},{{\theta }^{″}}_{13},{A^{″}}_1,{A^{″}}_2\right)$ 满足下列条件

$\begin{array}{l}{{\theta }^{\prime }}_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right){A^{\prime }}_1\left(v\right)\otimes w-{{\theta }^{\prime }}_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right){A^{\prime }}_1\left(v\right)\otimes {A^{″}}_1\left(w\right)\\ +v\otimes {{\theta }^{″}}_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right){A^{″}}_1\left(w\right)-{A^{\prime }}_1\left(v\right)\otimes {{\theta }^{″}}_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right){A^{″}}_1\left(w\right)\\ +{{\theta }^{\prime }}_{12}\left({\alpha }_2\left(x\right),{\alpha }_1\left(u\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left(z,y\right)\left(w\right)+{{\theta }^{\prime }}_{12}\left(z,y\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_2\left(x\right),{\alpha }_1\left(u\right)\right)\left(w\right)\\ -{{\theta }^{\prime }}_{13}\left(y,u\right)\left(v\right)\otimes {{\theta }^{″}}_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(x\right)\right)\left(w\right)-{{\theta }^{\prime }}_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(x\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{13}\left(y,u\right)\left(w\right)\\ +{{\theta }^{\prime }}_{12}\left(x,y\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\left(w\right)+{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{13}\left(x,y\right)\left(w\right)=0,\end{array}$ (1.21)

$\begin{array}{l}{{\theta }^{\prime }}_{13}\left({\alpha }_1\left(x\right),\left\{z,u,y\right\}\right){A^{\prime }}_2\left(v\right)\otimes w-{{\theta }^{\prime }}_{13}\left({\alpha }_1\left(x\right),\left\{z,u,y\right\}\right){A^{\prime }}_2\left(v\right)\otimes {A^{″}}_2\left(w\right)\\ +v\otimes {{\theta }^{″}}_{13}\left({\alpha }_1\left(x\right),\left\{z,u,y\right\}\right){A^{″}}_2\left(w\right)-{A^{\prime }}_2\left(v\right)\otimes {{\theta }^{″}}_{13}\left({\alpha }_1\left(x\right),\left\{z,u,y\right\}\right){A^{″}}_2\left(w\right)\\ +v\otimes {{\theta }^{″}}_{13}\left({\alpha }_1\left(x\right),\left\{z,u,y\right\}\right){A^{″}}_2\left(w\right)-{A^{\prime }}_2\left(v\right)\otimes {{\theta }^{″}}_{13}\left({\alpha }_1\left(x\right),\left\{z,u,y\right\}\right){A^{″}}_2\left(w\right)\\ +{{\theta }^{\prime }}_{13}\left(x,z\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_2\left(y\right),{\alpha }_1\left(u\right)\right)\left(w\right)+{{\theta }^{\prime }}_{12}\left({\alpha }_2\left(y\right),{\alpha }_1\left(u\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{13}\left(x,z\right)\left(w\right)\\ +{{\theta }^{\prime }}_{13}\left(x,y\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\left(w\right)+{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{13}\left(x,y\right)\left(w\right)=0,\end{array}$ (1.22)

$\begin{array}{l}{{\theta }^{\prime }}_{12}\left({\alpha }_2\left(z\right),{\alpha }_1\left(u\right)\right){{\theta }^{\prime }}_{12}\left(x,y\right)\left(v\right)\otimes w+{{\theta }^{\prime }}_{12}\left(x,y\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_2\left(z\right),{\alpha }_1\left(u\right)\right)\left(w\right)\\ +{{\theta }^{\prime }}_{12}\left({\alpha }_2\left(z\right),{\alpha }_1\left(u\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left(x,y\right)\left(w\right)+v\otimes {{\theta }^{″}}_{12}\left({\alpha }_2\left(z\right),{\alpha }_1\left(u\right)\right){{\theta }^{″}}_{12}\left(x,y\right)\left(w\right)\\ -{{\theta }^{\prime }}_{12}({\alpha }_2\left(z\right),\left\{y,x,u\right\}{A^{\prime }}_1\left(v\right)\otimes {A^{″}}_1\left(w\right)-{A^{\prime }}_1\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_2\left(z\right),\left\{y,x,u\right\}\right){A^{″}}_1\left(w\right)\\ -{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){{\theta }^{\prime }}_{12}\left(z,u\right)\left(v\right)\otimes w-{{\theta }^{\prime }}_{12}\left(z,u\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)w\\ -{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left(z,u\right)\left(w\right)-v\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){{\theta }^{″}}_{12}\left(z,u\right)w\\ +{{\theta }^{\prime }}_{12}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right){A^{\prime }}_1\left(v\right)\otimes {A^{″}}_1\left(w\right)+{A^{\prime }}_1\left(v\right)\otimes {{\theta }^{″}}_{12}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right){A^{″}}_1\left(w\right)=0,\end{array}$ (1.23)

$\begin{array}{l}{{\theta }^{\prime }}_{13}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right){A^{\prime }}_1\left(v\right)\otimes {A^{″}}_1\left(w\right)+{A^{\prime }}_1\left(v\right)\otimes {{\theta }^{″}}_{13}\left(\left\{x,y,z\right\},{\alpha }_2\left(u\right)\right){A^{″}}_1\left(w\right)\\ -{{\theta }^{\prime }}_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){{\theta }^{\prime }}_{12}\left(y,x\right)\left(v\right)\otimes \left(w\right)-{{\theta }^{\prime }}_{12}\left(y,x\right)\left(v\right)\otimes {{\theta }^{″}}_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\left(w\right)\\ -{{\theta }^{\prime }}_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left(y,x\right)\left(w\right)-v\otimes {{\theta }^{″}}_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){{\theta }^{″}}_{12}\left(y,x\right)\left(w\right)\\ -{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){{\theta }^{\prime }}_{13}\left(z,u\right)\left(v\right)\otimes w-{{\theta }^{\prime }}_{13}\left(z,u\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)\left(w\right)\\ -{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{13}\left(z,u\right)\left(w\right)-v\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){{\theta }^{″}}_{13}\left(z,u\right)\left(w\right)\\ +{{\theta }^{\prime }}_{13}\left({\alpha }_1\left(z\right),\left\{x,y,u\right\}\right){A^{\prime }}_1\left(v\right)\otimes {A^{″}}_2\left(w\right)+{A^{\prime }}_2(v)\otimes {{\theta }^{″}}_{13}\left({\alpha }_1\left(z\right),\left\{x,y,u\right\}\right){A^{″}}_2\left(w\right)=0,\end{array}$ (1.24)

$\begin{array}{l}{{\theta }^{\prime }}_{12}\left(\left\{x,y,z\right\},{\alpha }_1\left(u\right)\right){A^{\prime }}_2\left(v\right)\otimes {A^{″}}_2\left(w\right)+{A^{\prime }}_2\left(v\right)\otimes {{\theta }^{″}}_{12}\left(\left\{x,y,z\right\},{\alpha }_1\left(u\right)\right){A^{″}}_2\left(w\right)\\ -{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(z\right),\left\{y,x,u\right\}\right){A^{\prime }}_2\left(v\right)\otimes {A^{″}}_2\left(w\right)-{A^{\prime }}_2\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(z\right),\left\{y,x,u\right\}\right){A^{″}}_2\left(w\right)\\ -{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){{\theta }^{\prime }}_{12}\left(z,u\right)\left(v\right)\otimes w-{{\theta }^{\prime }}_{12}(z,u)(v)\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)\left(w\right)\\ -{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left(z,u\right)w-v\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right){{\theta }^{″}}_{12}\left(z,u\right)\left(w\right)\\ +{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){{\theta }^{\prime }}_{12}\left(x,y\right)\left(v\right)\otimes w+{{\theta }^{\prime }}_{12}\left(x,y\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\left(w\right)\\ +{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left(x,y\right)\left(w\right)+v\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){{\theta }^{″}}_{12}\left(x,y\right)\left(w\right)=0,\end{array}$ (1.25)

其中 $\forall x,y,z,u\in J,v\in V,w\in W$。

证明：由 ${{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12},{{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13},{A^{\prime }}_1\otimes {A^{″}}_1,{A^{\prime }}_2\otimes {A^{″}}_2$ 的定义知为线性映射。 $\left(V\otimes W,{{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12},{{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13},{A^{\prime }}_1\otimes {A^{″}}_1,{A^{\prime }}_2\otimes {A^{″}}_2\right)$ 是 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示当且仅当对于 $\left(V\otimes W,{{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12},{{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13},{A^{\prime }}_1\otimes {A^{″}}_1,{A^{\prime }}_2\otimes {A^{″}}_2\right)$ 有(1.3)~(1.8)成立。由于

$\begin{array}{l}{{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13}\left(x,y\right)\left(v\otimes w\right)-{{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13}\left(y,x\right)\left(v\otimes w\right)\\ ={{\theta }^{\prime }}_{13}\left(x,y\right)v\otimes w-{{\theta }^{\prime }}_{13}\left(y,x\right)v\otimes w+v\otimes {{\theta }^{″}}_{13}\left(x,y\right)w-v\otimes {{\theta }^{″}}_{13}\left(y,x\right)w,\end{array}$

则 ${{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13}$ 满足等式(1.3)当且仅当 ${{\theta }^{\prime }}_{13},{{\theta }^{″}}_{13}$ 满足等式(1.3)。在 $V\otimes W$ 上等式(1.4)左边

$\begin{array}{l}{{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12}\left({\alpha }_2\left(x\right),{\alpha }_1\left(u\right)\right){{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12}\left(z,y\right)\left(v\otimes w\right)-{{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(x\right)\right){{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13}\left(y,u\right)\left(v\otimes w\right)\\ -{{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right){A^{\prime }}_1\otimes {A^{″}}_1\left(v\otimes w\right)+{{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right){{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12}\left(x,y\right)\left(v\otimes w\right)\\ ={{\theta }^{\prime }}_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right){A^{\prime }}_1\left(v\right)\otimes w-{{\theta }^{\prime }}_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right){A^{\prime }}_1\left(v\right)\otimes {A^{″}}_1\left(w\right)\\ +v\otimes {{\theta }^{″}}_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right){A^{″}}_1\left(w\right)-{A^{\prime }}_1\left(v\right)\otimes {{\theta }^{″}}_{12}\left(\left\{z,u,x\right\},{\alpha }_2\left(y\right)\right){A^{″}}_1(\; w\; )\end{array}$

$\begin{array}{l}+{{\theta }^{\prime }}_{12}\left({\alpha }_2\left(x\right),{\alpha }_1\left(u\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left(z,y\right)\left(w\right)+{{\theta }^{\prime }}_{12}\left(z,y\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_2\left(x\right),{\alpha }_1\left(u\right)\right)\left(w\right)\\ -{{\theta }^{\prime }}_{13}\left(y,u\right)\left(v\right)\otimes {{\theta }^{″}}_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(x\right)\right)\left(w\right)-{{\theta }^{\prime }}_{13}\left({\alpha }_1\left(z\right),{\alpha }_2\left(x\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{13}\left(y,u\right)\left(w\right)\\ +{{\theta }^{\prime }}_{12}\left(x,y\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\left(w\right)+{{\theta }^{\prime }}_{12}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\left(v\right)\otimes {{\theta }^{″}}_{12}\left(x,y\right)\left(w\right),\end{array}$

因此 $\left(V\otimes W,{{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12},{{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13},{A^{\prime }}_1\otimes {A^{″}}_1,{A^{\prime }}_2\otimes {A^{″}}_2\right)$ 满足等式(1.4)当且仅当等式(1.21)式成立。同理， $\left(V\otimes W,{{\theta }^{\prime }}_{12}\otimes {{\theta }^{″}}_{12},{{\theta }^{\prime }}_{13}\otimes {{\theta }^{″}}_{13},{A^{\prime }}_1\otimes {A^{″}}_1,{A^{\prime }}_2\otimes {A^{″}}_2\right)$ 满足等式(1.5)当且仅当等式(1.22)成立，满足等式(1.6)当且仅当

等式(1.23)成立，满足等式(1.7)当且仅当等式(1.24)成立，满足等式(1.8)当且仅当等式(1.25)成立，因此，结论成立。

3. Hom-约当三系的扩张

定理2.1设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系， $\phi :J\times J\times J\to F$ 是三线性函数，在

$J\oplus V=\left\{x_1+{\lambda }_{1}c|x_1\in J,{\lambda }_1\in F\right\}$ 上定义

$\left\{x_1+{\lambda }_{1}c,x_2+{\lambda }_{2}c,x_3+{\lambda }_{3}c\right\}=\left\{x_1,x_2,x_3\right\}+\phi \left(x_1,x_2,x_3\right)c,$

$\left({\alpha }_1+id\right)\left(x_1+{\lambda }_{1}c\right)={\alpha }_1\left(x_1\right)+{\lambda }_{1}c,$

$\left({\alpha }_2+id\right)\left(x_1+{\lambda }_{1}c\right)={\alpha }_2\left(x_1\right)+{\lambda }_{1}c,$

则 $\left(J\oplus V,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1+id,{\alpha }_2+id\right)$ 为Hom-约当三系的充分必要条件是 $\phi$ 满足

$\phi \left(x_1,x_2,x_3\right)=\phi \left(x_3,x_2,x_1\right),$ (2.1)

$\begin{array}{l}\phi \left({\alpha }_1\left(x_1\right),{\alpha }_2\left(x_2\right),\left\{x_3,x_4,x_5\right\}\right)-\phi \left(\left\{x_1,x_2,x_3\right\},{\alpha }_1\left(x_4\right),{\alpha }_2\left(x_5\right)\right)\\ +\phi \left({\alpha }_1\left(x_3\right),\left\{x_2,x_1,x_4\right\},{\alpha }_2\left(x_5\right)\right)-\phi \left({\alpha }_1\left(x_3\right),{\alpha }_2\left(x_4\right),\left\{x_1,x_2,x_5\right\}\right)=0,\end{array}$ (2.2)

其中 $\forall x_1,x_2,x_3,x_4,x_5\in J,{\lambda }_1,{\lambda }_2,{\lambda }_3\in F$。

证明： $\forall x_1,x_2,x_3,x_4,x_5\in J,{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4,{\lambda }_5\in F$，则 $\left(J\oplus V,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1+id,{\alpha }_2+id\right)$ 为Hom-约当三系当且仅当下列等式成立

$\left\{x_1+{\lambda }_{1}c,x_2+{\lambda }_{2}c,x_3+{\lambda }_{3}c\right\}=\left\{x_3+{\lambda }_{3}c,x_2+{\lambda }_{2}c,x_1+{\lambda }_{1}c\right\},$ (2.3)

$\begin{array}{l}\left\{\left({\alpha }_1+id\right)\left(x_1+{\lambda }_{1}c\right),\left({\alpha }_2+id\right)\left(x_2+{\lambda }_{2}c\right),\left\{x_3+{\lambda }_{3}c,x_4+{\lambda }_{4}c,x_5+{\lambda }_{5}c\right\}\right\}\\ -\left\{\left({\alpha }_1+id\right)\left(x_3+{\lambda }_{3}c\right),\left({\alpha }_2+id\right)\left(x_4+{\lambda }_{4}c\right),\left\{x_1+{\lambda }_{1}c,x_2+{\lambda }_{2}c,x_5+{\lambda }_{5}c\right\}\right\}\\ -\left\{\left\{x_1+{\lambda }_{1}c,x_2+{\lambda }_{2}c,x_3+{\lambda }_{3}c\right\},\left({\alpha }_1+id\right)\left(x_4+{\lambda }_{4}c\right),\left({\alpha }_2+id\right)\left(x_5+{\lambda }_{5}c\right)\right\}\\ +\left\{\left({\alpha }_1+id\right)\left(x_3+{\lambda }_{3}c\right),\left\{x_2+{\lambda }_{2}c,x_1+{\lambda }_{1}c,x_4+{\lambda }_{4}c\right\},\left({\alpha }_2+id\right)\left(x_5+{\lambda }_{5}c\right)\right\}=0,\end{array}$ (2.4)

由 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系知 $\left\{x_1,x_2,x_3\right\}=\left\{x_3,x_2,x_1\right\}$，则等式(2.3)成立当且仅当 $\phi$ 满足等式(2.1)。由 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系知等式(1.2)成立，则等式(2.4)成立等价于等式(2.2)成立。因此，结论成立。

定理2.2设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系，且满足(1.11)~(1.15)， $\omega :J\times J\times J\to J^{*}$ 为三线性映射，在 $J\oplus J^{\text{*}}$ 上定义

$\left\{x+a^{*},y+b^{*},z+c^{*}\right\}=\left\{x,y,z\right\}+\omega \left(x,y,z\right)+L^{*}\left(x,y\right)c^{*}+L^{*}\left(z,y\right)a^{*}+M^{*}\left(x,z\right)b^{*},$

$\left({\alpha }_1+{\alpha }_1^{*}\right)\left(x+a^{*}\right)={\alpha }_1\left(x\right)+{\alpha }_1^{*}\left(a^{*}\right),$

$\left({\alpha }_2+{\alpha }_2^{*}\right)\left(x+a^{*}\right)={\alpha }_2\left(x\right)+{\alpha }_2^{*}\left(a^{*}\right),$

其中 $\forall x,y,z\in J,a^{*},b^{*},c^{*}\in J^{*}$，则 $\left(J\oplus J^{\text{*}},\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1+{\alpha }_1^{*},{\alpha }_2+{\alpha }_2^{*}\right)$ 为Hom-约当三系当且仅当 $\omega$ 满足

$\omega \left(x,y,z\right)=\omega \left(z,y,x\right),$ (2.5)

$\begin{array}{l}\omega \left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right),\left\{z,u,v\right\}\right)-\omega \left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right),\left\{x,y,v\right\}\right)\\ +\omega \left({\alpha }_1\left(z\right),\left\{y,x,u\right\},{\alpha }_2\left(v\right)\right)-\omega \left(\left\{x,y,z\right\},{\alpha }_1\left(u\right),{\alpha }_2\left(v\right)\right)\\ +L^{*}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)\omega \left(z,u,v\right)-L^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\omega \left(x,y,v\right)\\ -L^{*}\left({\alpha }_2\left(v\right),{\alpha }_1\left(u\right)\right)\omega \left(x,y,z\right)+M^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(v\right)\right)\omega \left(y,x,u\right)=0,\end{array}$ (2.6)

其中 $\forall x,y,z,u,v\in J$，此时称 $J\oplus J^{\text{*}}$ 为J的 $T^{*}$ -扩张。

证明：显然，在 $J\oplus J^{*}$ 上定义的新的运算对三个变量都是线性的。 $\forall x,y,z,u,v\in J$， $a^{*},b^{*},c^{*},d^{*},e^{*}\in J^{*}$， $\left(J\oplus J^{\text{*}},\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1+{\alpha }_1^{*},{\alpha }_2+{\alpha }_2^{*}\right)$ 为Hom-约当三系当且仅当在 $J\oplus J^{*}$ 上等式(1.1)、(1.2)成立。

由于

$\begin{array}{l}\left\{x+a^{*},y+b^{*},z+c^{*}\right\}-\left\{z+c^{*},y+b^{*},x+a^{*}\right\}\\ =\left\{x,y,z\right\}-\left\{z,y,x\right\}+\omega \left(x,y,z\right)-\omega \left(z,y,x\right)+L^{*}\left(x,y\right)c^{*}\\ -L^{*}\left(x,y\right)c^{*}+L^{*}\left(z,y\right)a^{*}-L^{*}\left(z,y\right)a^{*}+M^{*}\left(x,z\right)b^{*}-M^{*}\left(z,x\right)b^{*},\end{array}$

由于 $M^{*}\left(x,z\right)=M^{*}\left(z,x\right)$，因此在 $J\oplus J^{*}$ 上等式(1.1)成立等价于等式(2.5)成立。

在 $J\oplus J^{*}$ 上等式(1.2)左边直接计算得

$\begin{array}{l}\left\{\left({\alpha }_1+{\alpha }_1^{*}\right)\left(x+a^{*}\right),\left({\alpha }_2+{\alpha }_2^{*}\right)\left(y+b^{*}\right),\left\{z+c^{*},u+d^{*},v+e^{*}\right\}\right\}\\ -\left\{\left({\alpha }_1+{\alpha }_1^{*}\right)\left(z+c^{*}\right),\left({\alpha }_2+{\alpha }_2^{*}\right)\left(u+d^{*}\right),\left\{x+a^{*},y+b^{*},v+e^{*}\right\}\right\}\\ -\left\{\left\{x+a^{*},y+b^{*},z+c^{*}\right\},\left({\alpha }_1+{\alpha }_1^{*}\right)\left(u+d^{*}\right),\left({\alpha }_2+{\alpha }_2^{*}\right)\left(v+e^{*}\right)\right\}\\ +\left\{\left({\alpha }_1+{\alpha }_1^{*}\right)\left(z+c^{*}\right),\left\{y+b^{*},x+a^{*},u+d^{*}\right\},\left({\alpha }_2+{\alpha }_2^{*}\right)\left(v+e^{*}\right)\right\}\\ =\left\{{\alpha }_1\left(x\right),{\alpha }_2\left(y\right),\left\{z,u,v\right\}\right\}-\left\{{\alpha }_1\left(z\right),{\alpha }_2\left(u\right),\left\{x,y,v\right\}\right\}-\left\{\left\{x,y,z\right\},{\alpha }_1\left(u\right),{\alpha }_2\left(v\right)\right\}\end{array}$

$\begin{array}{l}+\left\{{\alpha }_1\left(z\right),\left\{y,x,u\right\},{\alpha }_2\left(v\right)\right\}+\omega \left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right),\left\{z,u,v\right\}\right)-\omega \left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right),\left\{x,y,v\right\}\right)\\ -\omega \left(\left\{x,y,z\right\},{\alpha }_1\left(u\right),{\alpha }_2\left(v\right)\right)+\omega \left({\alpha }_1\left(z\right),\left\{y,x,u\right\},{\alpha }_2\left(v\right)\right)+L^{*}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)\omega \left(z,u,v\right)\\ -L^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)\omega \left(x,y,v\right)-L^{*}\left({\alpha }_2\left(v\right),{\alpha }_1\left(u\right)\right)\omega \left(x,y,z\right)+M^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(v\right)\right)\omega \left(y,x,u\right)\\ +(L^{*}\left(\left\{z,u,v\right\},{\alpha }_2\left(y\right)\right){\alpha }_1^{*}-L^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)L^{*}\left(v,y\right)\\ -L^{*}\left({\alpha }_2\left(v\right),{\alpha }_1\left(u\right)\right)L^{\text{*}}\left(z,y\right)+M^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(v\right)\right)M^{*}\left(y,u\right))(\; a\; *\; )\end{array}$

$\begin{array}{l}+(M^{*}\left({\alpha }_1\left(x\right),\left\{z,u,v\right\}\right){\alpha }_2^{*}-L^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)M^{*}\left(x,v\right)\\ -L^{*}\left({\alpha }_2\left(v\right),{\alpha }_1\left(u\right)\right)M^{*}(x,z)+M^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(v\right)\right)L^{*}\left(u,x\right))\left(b^{*}\right)\\ +(L^{*}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)L^{*}\left(v,u\right)-L^{*}(\left\{x,y,v\right\},{\alpha }_2(u)){\alpha }_1^{*}\\ -L^{*}\left({\alpha }_2\left(v\right),{\alpha }_1\left(u\right)\right)L^{*}\left(x,y\right)+L^{*}\left({\alpha }_2\left(v\right),\left\{y,x,u\right\}\right){\alpha }_1^{*})(\; c\; *\; )\end{array}$

$\begin{array}{l}+(L^{*}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)M^{*}\left(z,v\right)-M^{*}\left({\alpha }_1\left(z\right),\left\{x,y,v\right\}\right){\alpha }_2^{*}\\ -M^{*}\left(\left\{x,y,z\right\},{\alpha }_2\left(v\right)\right){\alpha }_1^{*}+M^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(v\right)\right)L^{*}\left(y,x\right))\left(d^{*}\right)\\ +(L^{*}\left({\alpha }_1\left(x\right),{\alpha }_2\left(y\right)\right)L^{*}\left(z,u\right)-L^{*}\left({\alpha }_1\left(z\right),{\alpha }_2\left(u\right)\right)L^{*}\left(x,y\right)\\ -L^{*}\left(\left\{x,y,z\right\},{\alpha }_1\left(u\right)\right){\alpha }_2^{*}+L^{*}\left({\alpha }_1\left(z\right),\left\{y,x,u\right\}\right){\alpha }_2^{*})(\; e\; *\; )\end{array}$

由 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系，等式(1.2)成立。又由 $\left(J^{*},L^{*},M^{*},{\alpha }_1^{*},{\alpha }_2^{*}\right)$ 为 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 的表示知对于 $\left(J^{*},L^{*},M^{*},{\alpha }_1^{*},{\alpha }_2^{*}\right)$ 有(1.4)~(1.8)成立，则在 $J\oplus J^{*}$ 上等式(1.2)成立等价于等式(2.6)成立。因此结论成立。

定义2.1设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系，f为J上的双线性函数，若满足

$f\left(\left\{x,y,z\right\},u\right)=f\left(z,\left\{x,y,u\right\}\right)\left(\forall x,y,z,u\in J\right),$

$f\left(\left\{x,y,z\right\},u\right)=f\left(y,\left\{x,u,z\right\}\right)\left(\forall x,y,z,u\in J\right),$

则称f为J上的不变函数。

定义2.2设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系，若J上存在非退化的、不变的、对称的双线性函数f，则称 $\left(J,f\right)$ 为度量Hom-约当三系。

定理2.3设 $\left(J,\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1,{\alpha }_2\right)$ 为Hom-约当三系且满足

$\left\{x,y,z\right\}=\left\{x,z,y\right\}\left(\forall x,y,z\in J\right)$, (2.7)

三线性映射 $\omega :J\times J\times J\to J^{\text{*}}$ 满足(2.5)、(2.6)并且

$\omega \left(z,u,x\right)y=\omega \left(x,y,u\right)z=\omega \left(x,y,z\right)u\left(\forall x,y,z,u\in J\right),$ (2.8)

在 $J\oplus J^{*}$ 上定义

$q\left(x+a^{*},y+b^{*}\right)=a^{*}\left(y\right)+b^{*}\left(x\right)\left(\forall x,y\in J,a^{*},b^{*}\in J^{*}\right),$

则 $\left(J\oplus J^{*},q\right)$ 是度量Hom-约当三系。

证明：由 $\omega$ 满足等式(2.5)、(2.6)知 $\left(J\oplus J^{\text{*}},\left\{\cdot ,\cdot ,\cdot \right\},{\alpha }_1+{\alpha }_1^{*},{\alpha }_2+{\alpha }_2^{*}\right)$ 为Hom-约当三系。

对于 $y+b^{*}$，若对于 $\forall x+a^{*}\in J\oplus J^{*}$ 满足 $q\left(x+a^{*},y+b^{*}\right)=a^{*}\left(y\right)+b^{*}\left(x\right)=0$，取 $a^{*}=0$，则 $q\left(x,y+b^{*}\right)=b^{*}\left(x\right)=0$，因此 $b^{*}=0$。取 $x=0$，则 $q\left(a^{*},y+b^{*}\right)=a^{*}\left(y\right)=0$，因此 $y=0$，从而 $y+b^{*}=0$，q是非退化的。

因为 $q\left(x+a^{*},y+b^{*}\right)=a^{*}\left(y\right)+b^{*}\left(x\right)=q\left(y+b^{*},x+a^{*}\right)$，所以q是对称的。

$\forall x,y,z,u\in J,a^{*},b^{*},c^{*},d^{*}\in J^{*}$，直接计算得

$\begin{array}{l}q\left(\left\{x+a^{*},y+b^{*},z+c^{*}\right\},u+d^{*}\right)\\ =q\left(\left\{x,y,z\right\}+\omega \left(x,y,z\right)+L^{*}\left(x,y\right)c^{*}+L^{*}\left(z,y\right)a^{*}+M^{*}\left(x,z\right)b^{*},u+d^{*}\right)\\ =\omega \left(x,y,z\right)u+L^{*}\left(x,y\right)c^{*}\left(u\right)+L^{*}\left(z,y\right)a^{*}\left(u\right)+M^{*}\left(x,z\right)b^{*}\left(u\right)+d^{*}\left(\left\{x,y,z\right\}\right),\end{array}$

$\begin{array}{l}q\left(z+c^{*},\left\{x+a^{*},y+b^{*},u+d^{*}\right\}\right)\\ =q\left(z+c^{*},\left\{x,y,u\right\}+\omega \left(x,y,u\right)+L^{*}\left(x,y\right)d^{*}+L^{*}\left(u,y\right)a^{*}+M^{*}\left(x,u\right)b^{*}\right)\\ =\omega \left(x,y,u\right)z+L^{*}\left(x,y\right)d^{*}\left(z\right)+L^{*}\left(u,y\right)a^{*}\left(z\right)+M^{*}\left(x,u\right)b^{*}\left(z\right)+c^{*}\left(\left\{x,y,u\right\}\right),\end{array}$

由(2.7)知

$M^{*}\left(x,u\right)b^{*}\left(z\right)=\langle b^{*},\left\{x,z,u\right\}\rangle =M^{*}\left(x,z\right)b^{*}\left(u\right),$

此外，

$L^{*}\left(x,y\right)a^{*}\left(u\right)=L^{*}\left(u,y\right)a^{*}\left(z\right),L^{*}\left(x,y\right)c^{*}\left(u\right)=c^{*}\left(\left\{x,y,u\right\}\right),L^{*}\left(x,y\right)d^{*}\left(z\right)=d^{*}\left(\left\{x,y,z\right\}\right),$

因此，

$q\left(\left\{x+a^{*},y+b^{*},z+c^{*}\right\},u+d^{*}\right)=q\left(z+c^{*},\left\{x+a^{*},y+b^{*},u+d^{*}\right\}\right).$

$\begin{array}{l}q\left(\left\{x+a^{*},y+b^{*},z+c^{*}\right\},u+d^{*}\right)\\ =q\left(\left\{x,y,z\right\}+\omega \left(x,y,z\right)+L^{*}\left(x,y\right)c^{*}+L^{*}\left(z,y\right)a^{*}+M^{*}\left(x,z\right)b^{*},u+d^{*}\right)\\ =\omega \left(x,y,z\right)u+L^{*}\left(x,y\right)c^{*}\left(u\right)+L^{*}\left(z,y\right)a^{*}\left(u\right)+M^{*}\left(x,z\right)b^{*}\left(u\right)+d^{*}\left(\left\{x,y,z\right\}\right),\end{array}$

$\begin{array}{l}q\left(y+b^{*},\left\{z+c^{*}，u+d^{*},x+a^{*}\right\}\right)\\ =q\left(y+b^{*},\left\{z,u,x\right\}+\omega \left(z,u,x\right)+L^{*}\left(z,u\right)a^{*}+L^{*}\left(x,u\right)c^{*}+M^{*}\left(z,x\right)d^{*}\right)\\ =\omega \left(z,u,x\right)\left(y\right)+L^{*}\left(z,u\right)a^{*}\left(y\right)+L^{*}\left(x,u\right)c^{*}\left(y\right)+M^{*}\left(z,x\right)d^{*}\left(y\right)+b^{*}\left(\left\{z,u,x\right\}\right),\end{array}$

由(2.7)知

$L^{\text{*}}\left(z,u\right)a^{*}\left(y\right)=\langle a^{*},\left\{z,u,y\right\}\rangle =L^{*}\left(z,y\right)a^{*}\left(u\right),$

$L^{*}\left(x,u\right)c^{*}\left(y\right)=\langle c^{*},\left\{x,u,y\right\}\rangle =L^{*}\left(x,y\right)c^{*}\left(u\right),$

此外，

$b^{*}\left(\left\{z,u,x\right\}\right)=M^{*}\left(x,z\right)b^{*}\left(u\right),M^{*}\left(z,x\right)d^{*}\left(y\right)=d^{*}\left(\left\{x,y,z\right\}\right),$

因此，

$q\left(\left\{x+a^{*},y+b^{*},z+c^{*}\right\},u+d^{*}\right)=q\left(y+b^{*},\left\{z+c^{*},u+d^{*},x+a^{*}\right\}\right),$

则q是不变的，结论成立。

4. 结论

本文给出了Hom-约当三系的表示的定义及扩张的两种方法，作为约当三系的推广，还可以对Hom-约当三系进入深入研究，比如还可以研究它的上同调，并应用到其他领域。